Let us put
\(\overline{x}^N := \frac{1}{N} \sum _{j=1}^N x^j,\quad \text{and} \quad \overline{\alpha }^N := \frac{1}{N} \sum _{j=1}^N \alpha ^j.\) 
It was shown in [1]} that, if there is an optimal control \((\hat{\alpha }^{1,N},\dots , \hat{\alpha }^{N,N})\) , then it satisfies
\(\hat{\alpha }^{i,N}_t&=-\frac{1}{2 R} \left[ B Y^{i}_t + \bar{S} \left( 1 - \frac{\bar{R} }{R + \bar{R}} \right) \overline{X_t}^N+ \left( \bar{B} - \frac{\bar{R} }{R + \bar{R}} (B + \bar{B}) \right) \overline{Y_t}^{N} \right]\\& = \ell (Y_t^i, \overline{X}_t^N, \overline{Y}_t^N)\) 

for every \(i=1,\dots , N\) ,
where the processes \((X^{i,N}, Y^{i,N}, Z^{i,j,N})\)  satisfy the FBSDE system
[left=]align*
d Xit = [ A Xit + A XtN + B it + B tN ] dt +  dWit
d Yit
= - { 2 Q Xit + [ 2 Q -S22 (R + R) ] XtN
+ A Yit + [ A -S (B + B)2 (R + R) ] YtN } dt + k=1N Zi,kt dWkt
 Xi0 =x,   YiT = 2QTXTi + 2QTXNT.

On the other hand, the optimal control \(\hat{\alpha }\)  of the McKean–Vlasov control problem satisfies
\(\hat{\alpha }_t = -\frac{1}{2 R} \left[ B Y_t + \bar{S} \left( 1 - \frac{\bar{R} }{R + \bar{R}} \right)\mathbb {E}[X_t]+ \left( \bar{B} - \frac{\bar{R} }{R + \bar{R}} (B + \bar{B}) \right)\mathbb {E}[Y_t] \right]\) 
where \((X,Y,Z)\)  solves the McKean-Vlasov FBSDE
[left=]align
d Xt = { A Xt + AE[Xt]+ B t + B E[t] } dt +  dWt

d Yt = - { 2 Q Xt + [ 2 Q -S22 (R + R) ]E[Xt]
+ A Yt + [ A -S (B + B)2 (R + R) ]E[Yt] } dt + Zt dWt
 X0=x,   YT = 2QTXT + 2QT E[XT],

see [1]}.
Essentially, this is a fully coupled McKean-Vlasov FBSDE system with linear coefficients.
The existence of a unique solution of this equation is guaranteed e.g. by [3]} if \(T\)  is small enough.
Furthermore,
it follows by [4]} that if \(T\)  is sufficiently small, then the solution \(V\)  of Equation REF  exists, is Lipschitz–continuous in \((x,\mu )\)  and it satisfies \(V(t,\chi ,\mu ) = Y_t^{t,\chi ,\mu }\) , which is the solution of () such that \(X_t = \chi \)  and \(\mathcal {L}(\chi ) =\mu \) .
Therefore, the process \(X\)  satisfies the (decoupled) equation
\(dX_t &= \ell _1\big (X_t, V(t, X_t, \mathcal {L}(X_t)), \mathcal {L}(X_t, V(t, X_t, \mathcal {L}(X_t))) \big )\,dt +\sigma \,dW_t\\&= \overline{\ell }_1(t,X_t, \mathcal {L}(X_t)) + \sigma \,dW_t.\) 

Now, consider the interacting particle system
\(d\mathcal {X}^{i,N}_t = \overline{\ell }_1\big (\mathcal {X}_t^{i,N},L^N(\underline{\mathcal {X}}_t)) \big ) + \sigma \,d W_t^i\) 
which is well defined, since \(\overline{\ell }_1\)  is Lipschitz–continuous, as the composition of two Lipschitz–continuous functions.
It follows by [5]} that the sequence \((L^N(\underline{\mathcal {X}}))_N\)  satisfies the LDP with rate function \(\mathcal {J}\) .
It is checked exactly as in the proof of Theorem REF  that \(\mathcal {J}\)  has compact sublevel sets.
Thus, it remains to show that \(X^{i,N}\)  and \(\mathcal {X}^{i,N}\)  are exponentially close.
It follows from a similar argument as in Lemma REF  that
\(\sup _N\Big \Vert \sum _{i=1}^N|X^{i,N} - \mathcal {X}^{i,N}|\Big \Vert _{\mathcal {S}^{\infty }(\mathbb {R}^\ell )} + \sup _N\Big \Vert \sum _{i=1}^N|Y^{i,i}-V(\cdot ,\mathcal {X}^{i,N},L^N(\underline{\mathcal {X}}))|\Big \Vert _{\mathcal {S}^{\infty }(\mathbb {R}^\ell )}< \infty .\) 
Subsequently using Chebyshev's inequality and (REF ), we have
\(\mathbb {P}\Big (\sup _{t\in [0,T]}\mathcal {W}_2(L^N(\underline{X}_t), L^N(\underline{\mathcal {X}}_t)) >\varepsilon \Big ) &\le \mathbb {P}\Big ( \Big \lbrace \frac{1}{N}\sup _{t\in [0,T]}\sum _{i=1}^N|X^{i,N}_t - \mathcal {X}^{i}_t|^2\Big \rbrace ^{1/2} \ge \varepsilon \Big )\\&\le \mathbb {E}\Big [\exp \Big \lbrace  \Big (\sum _{i=1}^N\Vert X^{\underline{\alpha },i} - \mathcal {X}^{i}\Vert ^2_\infty \Big )\Big \rbrace  \Big ]e^{-\varepsilon ^2N^2}\\&\le C e^{-\varepsilon ^2 N^2}\) 

from which we deduce that
\(\lim _{N\rightarrow \infty }\frac{1}{N}\log \mathbb {P}\Big (\sup _{t \in [0,T]}\mathcal {W}_1(L^N(\underline{X}_t), L^N(\underline{\mathcal {X}}_t)) \ge \varepsilon \Big ) = -\infty .\) 
Therefore, it follows by [6]} that the sequence \((L^N(\underline{X}))\)  satisfies the LDP with rate function \(\mathcal {J}\) .
